Let $\ a,x,y,z>0$. Prove that: $\frac{a+y}{a+z}x+\frac{a+z}{a+x}y+\frac{a+x}{a+y}z\geq{x+y+z}\geq\frac{a+z}{a+x}x+\frac{a+x}{a+y}y+\frac{a+y}{a+z}z$

$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule: 1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students; 2) Each student received the maximum possible points in each problem or got $ 0$ in it; Lasha got the least number of points. What's the maximal number of points he could have? Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 \minus{} k$ points in it.

Can we find a subset $A$ of $\mathbb{N}$ containing exactly five numbers such that sum of any three elements of $A$ is a prime number? Justify your answer.

In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

In whatever follows $f$ denotes a differentiable function from $\mathbb{R}$ to $\mathbb{R}$. $f \circ f$ denotes the composition of $f(x)$. $\textbf{(a)}$ If $f\circ f(x) = f(x) \forall x \in \mathbb{R}$ then for all $x$, $f'(x) =$ or $f'(f(x)) =$. Fill in the blank and justify. $\textbf{(b)}$Assume that the range of $f$ is of the form $ \left(-\infty , +\infty \right), [a, \infty ),(- \infty , b], [a, b] $. Show that if $f \circ f = f$, then the range of $f$ is $\mathbb{R}$. [hide=Hint](Hint: Consider a maximal element in the range of f)[/hide] $\textbf{(c)}$ If $g$ satisfies $g \circ g \circ g = g$, then $g$ is onto. Prove that $g$ is either strictly increasing or strictly decreasing. Furthermore show that if $g$ is strictly increasing, then $g$ is unique.

Let be given a triangle $ ABC$ with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$. Six distinct points $ A_1$, $ A_2$, $ B_1$, $ B_2$, $ C_1$, $ C_2$ not coinciding with $ A$, $ B$, $ C$ are chosen so that $ A_1$, $ A_2$ lie on line $ BC$; $ B_1$, $ B_2$ lie on $ CA$ and $ C_1$, $ C_2$ lie on $ AB$. Let $ \alpha$, $ \beta$, $ \gamma$ three real numbers satisfy $ \overrightarrow{A_1A_2} \equal{} \frac {\alpha}{a}\overrightarrow{BC}$, $ \overrightarrow{B_1B_2} \equal{} \frac {\beta}{b}\overrightarrow{CA}$, $ \overrightarrow{C_1C_2} \equal{} \frac {\gamma}{c}\overrightarrow{AB}$. Let $ d_A$, $ d_B$, $ d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $ AB_1C_1$ and $ AB_2C_2$; $ BC_1A_1$ and $ BC_2A_2$; $ CA_1B_1$ and $ CA_2B_2$. Prove that $ d_A$, $ d_B$ and $ d_C$ are concurrent if and only if $ \alpha a \plus{} \beta b \plus{} \gamma c \neq 0$.

In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$ . Prove that line $XY$ passes through the midpoint of $AC$. (F. Nilov )

If we write the sequence $\text{AAABABBB}$ along the perimeter of a circle, then every word of the length $3$ consisting of letters $A$ and $B$ (i.e. $\text{AAA}$, $\text{AAB}$, $\text{ABA}$, $\text{BAB}$, $\text{ABB}$, $\text{BBB}$, $\text{BBA}$, $\text{BAA}$) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a $k$-element alphabet along the perimeter of a circle in such a way that every word of the length $l$ (i.e. an ordered $l$-tuple of letters) occurs exactly once on the perimeter.

Find the conditions that the functions $M(x, y)$ and $N (x, y)$ must satisfy in order that the differential equation $Mdx + Ndy =0$ shall have an integrating factor of the form $f(xy).$ You may assume that $M$ and $N$ have continuous partial derivatives of all orders.

Define a sequence {$a_n$}$^{\infty}_{n=1}$ by $a_1 = 4, a_2 = a_3 = (a^2 - 2)^2$ and $a_n = a_{n-1}.a_{n-2} - 2(a_{n-1} + a_{n-2}) - a_{n-3} + 8, n \ge 4$, where $a > 2$ is a natural number. Prove that for all $n$ the number $2 + \sqrt{a_n}$ is a perfect square.

Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation: $$ \operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC). $$

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.

Positive integers $(p,a,b,c)$ called [i]good quadruple[/i] if a) $p $ is odd prime, b) $a,b,c $ are distinct , c) $ab+1,bc+1$ and $ca+1$ are divisible by $p $. Prove that for all good quadruple $p+2\le \frac {a+b+c}{3} $, and show the equality case.

Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that $$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$ where $ A^* $ is the adjugate of $ A. $

Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$

For the give functions in $\mathbb{N}$: [b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$); [b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$. solve the equation $\phi(\sigma(2^x))=2^x$.

If the five-digit number $3AB7C$ is divisible by $4$ and $9$ and $A<B<C$, what is $A+B+C$? $\text{(A) }3\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }17\qquad\text{(E) }26$

In how many ways can $n$ identical balls be distributed to nine persons $A,B,C,D,E,F,G,H,I$ so that the number of balls recieved by $A$ is the same as the total number of balls recieved by $B,C,D,E$ together,.

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

On some cells of a $2n \times 2n$ board are placed white and black markers (at most one marker on every cell). We first remove all black markers which are in the same column with a white marker, then remove all white markers which are in the same row with a black one. Prove that either the number of remaining white markers or that of remaining black markers does not exceed $n^2$.

Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$

The sum of three numbers is $ 98$. The ratio of the first to the second is $ \frac {2}{3}$, and the ratio of the second to the third is $ \frac {5}{8}$. The second number is: $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33$

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.